The 20-year-old trained with the England squad on Sunday because of a calf injury to right-back Kyle Walker, but is not eligible to play for them just yet having represented Finland at the Under-21 level.
"He didn't take a lot of convincing," said Hodgson. "We had a problem with Kyle Walker and wanted a full complement in our training session, so we called Carl up. He was in my thoughts anyway.
"I invited him to train with us knowing full well he couldn't take part in this game, but it gave me the opportunity to sit down and ask him the question.
"I don't want to trick him to play for England in a few minutes in a game to block him, but he had to make a decision because we're very interested in him, even if the competition is tougher to get him into the England team rather than the Finland team. He made it very clear he wanted to play for England."
The FA will now press ahead to get clearance from FIFA for Jenkinson, whose mother is from Finland and father is from England.